HINT: <no title>
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Start your answer by writing out:
(x+⎯⎯⎯⎯⎯)(x+⎯⎯⎯⎯⎯). Then you must find two numbers which make the factorisation work out correctly.
STEP: Set up the binomials based on the x2 term
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We have the expression x2+10x+24 and we must factorise it.
To factorise an expression means to write the expression as a product. For example, if you factorise the number 10, you would write 2⋅5... 2 and 5 are factors of 10. In the case of a trinomial like x2+10x+24 factorising usually leads to two binomials: we want two binomials which have a product equal to x2+10x+24, just like 2 and 5 have a product of 10.
To work out the binomial factors requires patience, because
there is no single calculation which will give us the answer. Rather we
must take the time to work out the answer by trial and error (guess
& check).
The first step is to set up the answer based on the x2 term in the trinomial. It tells us that the binomials must be like this:
(x+⎯⎯⎯⎯⎯)(x+⎯⎯⎯⎯⎯)
Imagine starting FOIL with these two brackets: you will get x2 from the two x's, which is the first part of x2+10x+24.
STEP: Find the numbers which belong in the binomials
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So far we have this: x2+10x+24=(x+⎯⎯⎯⎯⎯)(x+⎯⎯⎯⎯⎯). Now we need to figure out which numbers belong in the empty spaces. Whatever numbers we try to put in the binomials, we must check the binomials with FOIL to make sure that everything works out correctly.
We should try numbers which have a product of 24. This is because of the "Lasts" term in FOIL. For example, let's check what happens if we try the numbers 2 and 12:
(x+2)(x+12)x2+12x+2x+24x2+14x+24⟵Use FOIL toexpand the product⟵Is this thetrinomial we want?
Shucks - we did not get the trinomial we needed because the x term is not correct. That means that the binomials (x+2) and (x+12) cannot be the correct factors.
This is where patience is required: we need to try two other numbers which have a product of 24
and see if those numbers work out to make the trinomial we need. We
must continue this process until we find two binomials which agree with
the trinomial. The correct pair of binomials is (x+4)(x+6) because (x+4)(x+6)=x2+10x+24.
The correct answer is: (x+4)(x+6).
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